REST AND MOTION

Motion is change in the position of an object with time. When do we say that a body is at rest and when do we say that it is in motion? You may say that if a body does not change its position as time passes it is at rest. If a body changes its position with time, it is said to be moving.  For example, A book placed on the table remains on the table and we say that the book is at rest (if it is viewed from the room). However, if we station ourselves on the moon the whole earth is found to be changing its position and so the room, the table, and the book are all continuously changing their positions. Therefore the book is moving if it is viewed from the moon. There is no meaning of rest or motion without the viewer. Nothing is in absolute rest or in absolute motion. The moon is moving with respect to the book and the book moves with respect to the moon.

To locate the position of a particle we need a frame of reference. A convenient way to fix up the frame of reference is to choose three mutually perpendicular axes and name them X-Y-Z axes. The coordinates, (x, y, z) of the particle then specify the position of the particle with respect to that frame. Add a clock into the frame of reference to measure the time. If all the three coordinates x, y, and z of the particle remain unchanged as time passes, we say that the particle is at rest with respect to this frame. If any one or more coordinates change with time, we say that the body is moving with respect to this frame.

DISTANCE AND DISPLACEMENT

The distance traversed by the car from Point A to point B is called the path length. Assume the car is at point A at time \(t_1\) and at point B at time \(t_2\) with respect to a given reference frame. During the time interval \(t_1\) to \(t_2\) the particle moves along the path ACB. The length of the path ACB is called the distance travelled during the time interval \(t_1\)to \(t_2\). 

Displacement is the change in position of an object. Let \(x_{1}\) and \(x_{2}\) be the positions of an object at time \(t_{1}\) and \(t_{2}\). Then its displacement, denoted by \(\Delta x\), in time \(\Delta t=\left(t_{2}-t_{1}\right)\), is given by the difference between the final and initial positions:
\(\Delta x=x_{2}-x_{1}\)
(We use the Greek letter delta \((\Delta)\) to denote a change in a quantity.)
If \(x_{2}>x_{1}, \Delta x\) is positive; and if \(x_{2}<x_{1}, \Delta x\) is negative. Displacement has both magnitude and direction. Such quantities are represented by vectors. 

You will read about vectors in the next chapter. Presently, we are dealing with motion along a straight line (also called rectilinear motion) only. In one-dimensional motion, there are only two directions (backward and forward, upward and downward) in which an object can move, and these two directions can easily be specified by + and – signs. For example, displacement of the car in moving from \(O\) to \(P\) is :
\(
\Delta x=x_2-x_1=(+360 \mathrm{~m})-0 \mathrm{~m}=+360 \mathrm{~m}
\)
The displacement has a magnitude of \(360 \mathrm{~m}\) and is directed in the positive \(x\) direction as indicated by the + sign. Similarly, the displacement of the car from \(P\) to \(Q\) is \(240 m-360 m=-120 \mathrm{~m}\). The negative sign indicates the direction of displacement. Thus, it is not necessary to use vector notation for discussing the motion of objects in one dimension.

The magnitude of displacement may or may not be equal to the path length traversed by an object. For example, for the motion of the car from \(\mathrm{O}\) to \(\mathrm{P}\), the path length is \(+360 \mathrm{~m}\) and the displacement is \(+360 \mathrm{~m}\). In this case, the magnitude of displacement \((360 \mathrm{~m})\) is equal to the path length \((360 \mathrm{~m})\). But consider the motion of the car from \(\mathrm{O}\) to \(\mathrm{P}\) and back to \(\mathrm{Q}\). In this case, the path length \(=(+360 \mathrm{~m})+(+120 \mathrm{~m})=+\) \(480 \mathrm{~m}\). However, the displacement \(=(+240 \mathrm{~m})-\) \((0 \mathrm{~m})=+240 \mathrm{~m}\). Thus, the magnitude of displacement \((240 \mathrm{~m})\) is not equal to the path length (480 m).

The magnitude of the displacement for a course of motion may be zero but the corresponding path length is not zero. For example, if the car starts from O, goes to \(\mathrm{P}\) and then returns to \(O\), the final position coincides with the initial position and the displacement is zero. However, the path length of this journey is \(\mathrm{OP}+\mathrm{PO}=360 \mathrm{~m}+360 \mathrm{~m}=720 \mathrm{~m}\)

Example: 3.1:An old person moves on a semi-circular track of radius \(60.0 \mathrm{~m}\) during a morning walk. If he starts at one end of the track and reaches at the other end of the track, find the distance covered and the displacement of the person.

Solution: The distance covered by the person equals the length of the track. It is equal to \(\pi R=\pi \times 60.0 \mathrm{~m}\) \(=188.49 \mathrm{~m}\). The displacement is equal to the diameter of the semi-circular track joining the two ends. It is \(2 R=2\) \(\times 60.0 \mathrm{~m}=120 \mathrm{~m}\).

Motion of an Object Position-time graph

The motion of an object can be represented by a position-time graph as shown in Fig 3.2. Such a graph is a powerful tool to represent and analyse different aspects of the motion of an object. For motion along a straight line, say \(X\)-axis, only \(x\)-coordinate varies with time and we have an \(x-t\) graph. Let us first consider the simple case in which an object is stationary, e.g. a car standing still at \(x=40 \mathrm{~m}\). The position-time graph is a straight line parallel to the time axis, as shown in Fig. 3.2(a).

If an object moving along the straight line covers equal distances in equal intervals of time, it is said to be in uniform motion along a straight line. Fig. 3.2(b) shows the position-time graph of such a motion.

Motion of a Car

Now, let us consider the motion of a car that starts from rest at time \(t=0 \mathrm{~s}\) from the origin \(O\) and picks up speed till \(t=10 \mathrm{~s}\) and thereafter moves with uniform speed till \(t=18 \mathrm{~s}\). Then the brakes are applied and the car stops at \(t=20 \mathrm{~s}\) and \(x=296 \mathrm{~m}\). The position-time graph for this case is shown in Fig. 3.3.